Homoclinism of Lie rings

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Definition

Short definition

A homoclinism of Lie rings is a homologism of Lie rings with respect to the subvariety of the variety of Lie rings given by abelian Lie rings.

Full definition

For any Lie ring L, let Z(L) denote the center of L, \operatorname{Inn}(L) denote the Lie ring of inner derivations of L (explicitly, it is isomorphic to L/Z(L), and L' denote the derived subring of L.

Let \gamma_L denote the mapping \operatorname{Inn}(L) \times \operatorname{Inn}(L) \to L' that arises from the Lie bracket mapping L \times L \to L', and then observing that this map is constant on the cosets of Z(L) \times Z(L). Note that the mapping is \mathbb{Z}-bilinear.

A homoclinism of Lie rings L and M is a pair (\zeta,\varphi) where \zeta is a homomorphism of \operatorname{Inn}(L) with <math\operatorname{Inn}(M)</math> and \varphi is a homomorphism of L' with M', such that \varphi \circ \gamma_L = \gamma_M \circ (\zeta \times \zeta).

Related notions

Term Meaning
category of Lie rings with homoclinisms category whose objects are Lie rings and whose morphisms are homoclinisms
isoclinism of Lie rings an invertible homoclinism, or equivalently, a homoclinism where both the component homomorphisms are isomorphisms
homoclinism of groups the analogous concept for groups
homologism of Lie rings a more general concept of which homoclinisms are a special case. Homoclinisms are homologisms with respect to the subvarety of abelian groups in the variety of Lie rings.